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2 kinds of integral

  1. Apr 10, 2008 #1
    Hi all,

    I need help.

    What is the difference between

    [tex]
    \int_{0}^b x^2 dx
    [/tex]

    with

    [tex]
    lim_{b\rightarrow \ \infty} \int_{0}^b x^2 dx
    [/tex]

    ???

    Can someone please show me algebraically for its clarity?

    I don’t understand of the 2nd integral equation means.
    Especially the appears of limit as b approach to infinity.

    Thanks in advance
     
  2. jcsd
  3. Apr 10, 2008 #2
    one is a proper integral and the other one is an improper integral
     
  4. Apr 10, 2008 #3
    The first integral is a regular definite integral. It basically just measures the area that the parabola [tex]y=x^2[/tex] encloses with the x-axis from the point x=0 to another point x=b.


    The second one is an improper integral. THati is:

    [tex]\int_{0}^{\infty}x^2dx=\lim_{b\to\infty}\int_{0}^{b}x^2dx[/tex]

    SO it means that the function x^2 is unbounde from the above. And if that integral converges, than it means that we are calculating the area that the function [tex]f(x)=x^2[/tex] encloses with the x-axis from x=0 to infinity. In other words the area that the right wing of the parabola encloses with the x-axis.
     
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